Conjecture on frieze counts for exceptional Dynkin types

A frieze of a Dynkin type assigns values to the nodes of its Dynkin diagram. A unitary frieze is obtained by evaluating all cluster variables in a given cluster at $1$.

Frieze-count conjecture. The numbers of friezes of types $E_6$, $E_7$, $E_8$ and $F_4$ are $868$, $4400$, $26592$ and $112$, respectively.

These counts concern the sporadic Dynkin types not covered by the preceding general formulas for types $A$, $B$, $C$, $D$ and $G$. The paper describes an algorithm for enumerating these friezes, based on a separate conjectural bound for frieze values.